querent

Spam yourself with math PhD qualifying-exam questions.

UC Irvine Spring 2026, 5 Real analysis, Egorov's theorem, measurable functions

Let ${(𝑓_{𝑘})}_{𝑘 \in ℕ}$ be a sequence of measurable real-valued functions defined on a measurable set $𝐸$ with

𝑚 (𝐸) < \infty .

Suppose that, for each $𝑥 \in 𝐸$ ,

| 𝑓_{𝑘} (𝑥) | \leq 𝑀_{𝑥} < \infty, 𝑘 \in ℕ

for some $𝑀_{𝑥} \geq 0$ . Show that for every $𝜀 > 0$ , there exist a closed set $𝐹 \subset 𝐸$ and a finite constant $𝑀$ such that

𝑚 (𝐸 \ 𝐹) \leq 𝜀

and

| 𝑓_{𝑘} (𝑥) | \leq 𝑀, 𝑘 \in ℕ, 𝑥 \in 𝐹 .

UC Irvine Spring 2026, 6 Real analysis, bounded linear operators, sequence spaces, operator norm

Consider a linear operator $𝑇 : ℓ^{2} (ℕ) \to ℝ$ defined by

𝑇 (\vec{𝑥}) = \sum_{𝑛 = 1}^{\infty} 2^{- 𝑛} 𝑥_{2 𝑛},

where

\vec{𝑥} = (𝑥_{1}, 𝑥_{2}, 𝑥_{3}, \dots) \in ℓ^{2} (ℕ) .

Is $𝑇$ bounded? If yes, find $∥ 𝑇 ∥$ .

UC Irvine Spring 2026, 6 Real analysis, absolute continuity, summation by parts, improper integrals

Let $𝑓 : [0, \infty) \to [0, \infty)$ be locally absolutely continuous with

𝑓^{'} \in 𝐿^{1} ([0, \infty)) .

Prove that the limit

lim_{𝑛 \to \infty} (\sum_{𝑘 = 1}^{𝑛} 𝑓 (𝑘) - \int_{0}^{𝑛} 𝑓 (𝑥) d 𝑥)

exists.

UC Irvine Spring 2026, 7 Real analysis, inverse function theorem, matrix equations

Show that there exist $𝜀 > 0$ and $𝛿 > 0$ such that for any $2 \times 2$ matrix $𝐴$ that is $𝛿$ -close to identity, there exists a unique $2 \times 2$ matrix $𝑋$ in the $𝜀$ -neighborhood of the identity matrix such that

2 𝑋^{3} - 𝑋^{2} = 𝐴 .

UC Irvine Spring 2026, 8 Real analysis, iterated integrals, Fubini's theorem

Evaluate the integral

\int_{0}^{2} (\int_{0}^{4 - 𝑥^{2}} \frac{𝑥 𝑒^{2 𝑦}}{4 - 𝑦} d 𝑦) d 𝑥 .

UC Irvine Spring 2026, 9 Real analysis, flux integrals, vector fields, surface integrals

A fisherman's net has a rim, which is a circle of radius $5$ . He fixes it in the sea in such a way that the rim is in the $𝑥 𝑧$ -plane with center at the origin. The velocity of water is given by the vector field $\vec{𝐹}$ given by

\vec{𝐹} (𝑥, 𝑦, 𝑧) = (9 𝑥 𝑦^{2} + 𝑧^{2}, 2 - 3 𝑦^{3}, 𝑥^{2}) .

Find the flux of the water across the net.

Stanford Fall 2023, 1 Topology, Lefschetz fixed point theorem, Euler characteristic

Let $𝑀$ be a closed, smooth, odd dimensional manifold, and $𝑓 : 𝑀 \to 𝑀$ a diffeomorphism with only non-degenerate fixed points. Show that the number of fixed points must be even.

Stanford Fall 2023, 1 Topology, tangent bundle, orientability

Let $𝑀$ be a smooth manifold. Show that the tangent bundle $𝑇 𝑀$ is always an orientable manifold.

Stanford Fall 2023, 2 Topology, orientability, Alexander duality, Mayer-Vietoris

Let $𝑀 \subset ℝ^{3}$ be a closed, connected, 2-dimensional submanifold.

(a) Show that $𝑀$ is orientable.

(b) Compute the homology groups $𝐻_{𝑖} (ℝ^{3} - 𝑀)$ in terms of the genus of $𝑀$ .

Stanford Fall 2023, 2 Topology, degree theory, mapping degree, homotopy

(a) Suppose $𝑋, 𝑌, 𝑍$ are closed oriented manifolds such that

dim (𝑋) = dim (𝑌) > dim (𝑍) .

Let $𝑔 : 𝑋 \to 𝑍$ and $ℎ : 𝑍 \to 𝑌$ be smooth maps. Show that $deg (ℎ \circ 𝑔) = 0$ .

(b) Find a map $𝑓 : 𝑇^{2} \to 𝑇^{2}$ with $deg (𝑓) = 0$ but so that $𝑓$ is not null-homotopic.

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